Why diversify your investments?

More baskets is better

Variance and standard deviation

• Variance is expected squared deviation from mean
  • Flipping once for $10 \(\Rightarrow\) variance = 100
  • Flipping twice for $5 \(\Rightarrow\) variance = (1/4) \(\times\) 100 + (1/4) \(\times\) 100 = 50
• Standard deviation is square root of variance
  • Flipping once for $10 \(\Rightarrow\) std dev = 10
  • Flipping twice for $5 \(\Rightarrow\) std dev = 7.07

Investment interpretation

• Suppose we start with $100 before we flip
• There are two independent assets, each with possible returns of \(\pm\) 10%
  • Put all $100 in a single asset \(\leftrightarrow\) flip once for $10
  • Put $50 in each asset \(\leftrightarrow\) flip twice for $5
• Call the assets 1 and 2 with random returns \(r_1\) and \(r_2\)
• The portfolio return is less risky than either \(r_1\) or \(r_2\).

Portfolio Returns

• Consider a $100,000 portfolio with 40% in one asset (asset 1) and 60% in a second (asset 2).
• Suppose asset 1 \(\uparrow\) 20%. $40,000 \(\rightarrow\) $48,000
• Suppose asset 2 \(\uparrow\) 10%. $60,000 \(\rightarrow\) $66,000
• $100,000 \(\rightarrow\) $114,000
• So, up 14% = 0.4 \(\times\) 20% + 0.6 \(\times\) 10%
• The portfolio return is \(r_p = w_1r_1 + w_2r_2\) where \(w_i\) is the fraction of the portfolio invested in asset \(i\).

Expected portfolio return

• \(\bar{r}_1\) = expected return of asset 1
• \(\bar{r}_2\) = expected return of asset 2
• The expected portfolio return is

Portfolio variance

• The deviation of the portfolio return from its mean is